
# q-expansion of newform 175.3.d.f, downloaded from the LMFDB on 25 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 175
weight = 3
poly_data = [5, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0], 1], [[0, 2], 1]]

hecke_ring_character_values = [[127, [1, 0]], [101, [-1, 0]]]
aps_data = [[1, 0], [0, 1], [0, 0], [-7, 0], [2, 0], [0, -3], [0, 6], [0, 3], [-26, 0], [-22, 0], [0, 12], [-14, 0], [0, 6], [34, 0], [0, -6], [34, 0], [0, 9], [0, -21], [-14, 0], [62, 0], [0, -12], [38, 0], [0, 9], [0, 6], [0, -6], [0, -15], [0, 36], [106, 0], [-142, 0], [34, 0], [-194, 0], [0, 27], [166, 0], [0, 21], [-142, 0], [2, 0], [0, -15], [34, 0], [0, -24], [0, 33], [218, 0], [0, 57], [-58, 0], [-206, 0], [226, 0], [0, -30], [-118, 0], [0, 18], [0, 57], [0, -3], [214, 0], [98, 0], [0, 36], [0, -75], [0, -30], [34, 0], [0, -57], [0, -72], [-14, 0], [2, 0], [0, -21], [0, 75], [0, 45], [0, 114], [0, 72], [-374, 0], [482, 0], [-494, 0], [346, 0], [0, 75], [0, 6], [338, 0], [0, 66], [-86, 0], [-262, 0], [0, -126], [698, 0], [0, 69], [-538, 0], [0, -66], [0, 183], [-118, 0], [-718, 0], [0, 114], [0, 6], [634, 0], [338, 0], [466, 0], [0, 99], [-206, 0], [0, 81], [0, -48], [166, 0], [-838, 0], [-262, 0], [0, 96], [0, -201], [0, -162], [0, -117], [842, 0], [-134, 0], [706, 0], [0, -3], [-82, 0], [-118, 0], [0, -198], [0, -177], [0, -30], [398, 0], [0, -30], [0, -210], [-206, 0], [-494, 0], [0, -27], [542, 0], [-298, 0], [0, 225], [0, 144], [154, 0], [338, 0], [0, 129], [814, 0], [0, -153], [-926, 0], [0, -129], [362, 0], [1058, 0], [0, 108], [0, 252], [0, -291], [-982, 0], [694, 0], [242, 0], [106, 0], [0, -246], [0, -252], [0, 183], [0, 153], [0, 69], [1358, 0], [0, 69], [482, 0], [-926, 0], [226, 0], [0, 327], [0, 186], [0, -9], [0, 60], [0, 69], [514, 0], [1306, 0], [0, 252], [-1526, 0], [0, 348], [-734, 0], [1202, 0], [-1282, 0], [0, 252], [0, 48], [0, -189], [-734, 0], [934, 0], [-314, 0], [0, 33], [1486, 0], [0, -216], [-58, 0], [0, -141]]
